Semistable reduction for overconvergent F-isocrystals, I: Unipotence and logarithmic extensions

Let X be a smooth variety over a field of positive characteristic, and let E be an overconvergent isocrystal on X. We establish a criterion for the existence of a “canonical logarithmic extension” of E to a good compactification of X. In the process, we construct a category of overconvergent log-isocrystals and discuss its basic properties. In subsequent work, we will use these results to show that a canonical logarithmic extension always exists after pulling back E along a suitable cover of X.